## Motion in Two Dimensions

Motion in two dimensions is one of the most common phenomena that we come across in our everyday life. Depending upon the path taken by an object, a motion can be of different types like projectile motion, rectilinear motion, rotational motion, etc. By motion in two dimensions, we mean that the position of an object can vary both in X and Y axes simultaneously. Therefore, its position at any instant of time can be specified using (x,y) coordinates.

We are familiar with the equations of motion in one dimension. These equations can also be extended to two dimensions by separately applying these equations to both the X and Y axes. For Y-axis

\(v_y ~ = ~u_y~+ ~a_y t\)

\(s_y~ =~ u_y t~ +~ \frac12 a_y t^2\)

\(v_y^2\) =\( u_y^2~ +~ 2a_y s\)

Where,

\(v_y\) = final velocity of object in y direction

\(u_y\) = initial velocity of object in y direction

\(s_y\) = displacement of object in y direction

\(a_y\) = acceleration of object in y direction

\(t\) = time interval in which object is under consideration

Similarly, for **X-axis**

\(v_x\) = \( u_x~+~ a_x t\)

\(s_x\) = \(u_x t ~+~ \frac12 a_x t^2\)

\(v_x^2\)= \(u_x^2 + 2a_x s\)

Where,

\(v_x\) = final velocity of object in x direction

\(u_x\) = initial velocity of object in x direction

\(s_x\) = displacement of object in x direction

\(a_x\) = acceleration of object in x direction

\(t\) = time interval in which object is under consideration

When we say motion of body A relative to B we mean motion of A, as observed from B’s frame of reference. Mathematically it is represented as:

\(V_{BA} ~= ~V_B ~ – ~V_A\)

Where, \(V_{BA}\) = Velocity of B as observed from A

\(V_B\) = Velocity of B from earth as reference frame

\(V_A \) = Velocity of A from earth as reference frame

Consider a man running uphill while it is raining. It is difficult to decide from which side the man will get wet if we do not know the relative m of the man and rain with respect to each other. Suppose, the components of velocity of man are given as 6i + j and that of rain is given by 3i – 3j. If we can find the angle at which the rain hits the man, we can easily decide from which direction the man gets hit by the rain. To find the relative velocity or relative motion of rain with respect to man, the velocity of man with respect to rain must be subtracted, i.e.,

\( \overrightarrow{v_{r|m}}~ =~\overrightarrow{v_{r}} ~-~\overrightarrow{v_{m}} \)

Relative motion can be explained using the vector representation as:

It can be seen that with respect to the man, the rain is falling at an angle of \( tan^{-1} \left( \frac 34 \right) \). So, if the man is running in positive X- direction then rain must hit him from the front direction. So, if the man has to wear a bag he must wear it on his back to prevent it from rain. Thus only by knowing the velocities, the direction of hitting of rain can be determined. Now, what happens if the horizontal component of the velocity of rain and man becomes equal? Interesting, isn’t it? Well, in such a situation the rain won’t hit the man from either side. So, to answer the question whether one should run fast or slow in the rain to get less wet, it all depends on the **velocities **or** relative motion**.